Yes, the Bible does say that God not only forgives our sins when we repent and trust Christ for our salvation, but He also forgets them. The Bible says, "I will forgive their wickedness and will remember their sins no more" (Jeremiah 31:34).
How is this possible? On a human level it isn't, of course; we may remember what someone did to hurt us as long as we live (unless disease robs us of our memories). But with God it is possible; He is able to blot out our sins so completely that it is as if they had never existed. The Psalmist declared, "As far as the east is from the west, so far has he removed our transgressions from us" (Psalm 103:12).
Does this mean God just ignores our sins? No, not at all. God is holy and just, and sin is an offense to Him. When we sin, we're telling God that He doesn't matter to us; in fact, we're actually in open rebellion against Him. We are guilty and deserve only His judgment, and no matter how hard we try, we can't erase our sins or cleanse them from our hearts.
Only God can do this -- and that's exactly what He did when He sent His Son, Jesus Christ, to earth to die for us. We deserve to die -- but on the cross all our sins were placed on Him, and He died in our place. Do you know your sins have been forgiven? You can, by turning to Christ in repentance and faith, and giving your life to Him.